As we know, the DNA has more stability than RNA, if one chain has broken or accidentally distorted, it can be repaired by the other one.

Suppose there is a segment AGTC, its peer is GACT. Now its peer get distorted to TACT, so the DNA will replace the wrong T to G. But why doesn't the DNA replace the A to C in the first chain? How does the DNA know the first chain is correct and the second one is incorrect?

2 Answers
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The reason the fact it isn't realistic is important. DNA repair machinery works by repairing common errors that occur due to common mutational pathways. The enzymes are specific for this, for example one particular enzyme targets mutations caused by UV and itself is activated by sunlight thus it's specificity makes it repair the correct chain. Also during replication, newly synthesised DNA lacks methylation. The repair enzymes thus know which chain is correct.

Your specific scenario is not realistic, thymine is a pyrimidine base while guanine is a purine base. They are very different chemically and can't spontaneously convert into each other. So this is a type of damage that only occurs during the replication of DNA.

The typical damage that occurs and can be repaired is oxidation, methylation and hydrolysis of DNA. This kind of damage results in modified or missing bases and can be detected due to that.